For example, you can't integrate $2x\sqrt{1+ x^2}$ just using the two parts of the fundamental theorem of calculus. I can see that I can't solve it, but I'm not sure which unique characteristic of this function it is that makes it impossible to solve without using the substitution rule.
Is it just that it involves a product of two terms, both with $x$ in them? I don't think that's right because, for example, we can integrate $\tan x\sec x$ just fine.
So what characteristic of a function makes it impossible (or very difficult) to solve without using substitution?
I'm just trying to figure out why I can't solve this without using substitution. Any help is greatly appreciated.
As long as we're limited to one-dimensional integrals, substitution is never necessary. All you have to do to prove a result of the form $\int f(x)dx=g(x)+c$, or one of the form $\int_a^b f(x)dx=g(b)-g(a)$, is verify $g'=f$. For example, I can just announce $\int x\sqrt{1+x^2}dx=\frac{1}{3}(1+x^2)^{3/2}+C$, then differentiate the right-hand side to be sure. The reason we use substitution is because, human psychology being what it is, it makes antiderivatives easier to spot. So if you ask when that kind of need exists, that's really a psychological question.
When we get to multiple integrals, it gets more interesting. How do you evaluate $\int_{\Bbb R^2}\exp (-x^2-y^2) dx dy$? The change to polar coordinates introduces a Jacobian factor into the integrand, in a multivariable generalisation of one-dimensional substitution. In this case, there isn't really an antiderivative whose change from some point to some other point gives the integral by FTC. The double integral can be written as a product of two one-variable definite integrals, whether we work in Cartesian or polar coordinates; but regions of the form $x\in X\land y\in Y$ don't transform into those of the form $r\in R\land\theta\in\Theta$, or vice versa. By contrast, a membership equivalence description exists for any one-variable monotonic substitution.
That's not to say, however, which problems need to be discussed with multiple integrals. The double integral above arises when we try to evaluate $\int_{\Bbb R}\exp -x^2 dx$ by squaring it, but this single integral can be evaluated by other methods (see e.g. here). In that case, we can again do without any kind of substitution... with some effort.